Advanced Array Signal Processing Techniques for Beamforming and...

Fakultät für Elektrotechnik und Informationstechnik

In Zusammenarbeit mit

Advanced Array Signal Processing Techniques forBeamforming and Direction Finding

Masterarbeitzur Erlangung des akademischen Grades Master of Science

vorgelegt von: Jens Steinwandt

Matrikelnummer: 39733

geboren: 14. März 1986 in Leinefelde

Studiengang: Elektrotechnik und Informationstechnik

Studienrichtung: Informationstechnik und Kommunikationstechnik

Anfertigung im Fachgebiet: Nachrichtentechnik

Fakultät für Elektrotechnik und Informationstechnik

verantwortlicher Professor: Univ.-Prof. Dr.-Ing. Martin Haardt

Erster wissenschaftlicher Betreuer: Dr. Rodrigo C. de Lamare

Zweiter wissenschaftlicher Betreuer: Univ.-Prof. Dr.-Ing. Martin Haardt

Datum: 9. Mai 2011

Acknowledgements i

ACKNOWLEDGEMENTS

Firstly, I would like to express my sincere gratitude to Prof. Martin Haardt for giving

me the chance to work on such an interesting topic under his supervision. His constant

support of fresh ideas, comments and questions have encouraged me to go beyond what

I believed would be my limits.

Secondly, I am very grateful to Dr. Rodrigo C. de Lamare for his help, valuable

supervision and useful advice for my research, without which most of this work would

not have been possible.

Further thanks go to all members of the Communications Research Group at the

University of York and the members of the Communications Research Laboratory at

Ilmenau University of Technology, for their support throughout my master’s research.

I would like to thank my family, especially my mother, for constantly supporting

me in many ways over the last year in York. I would not have made it to this point if

you had not believed in me and encouraged me to discover my full potential.

Finally, I want to thank Lola for being there for me and for being the source of my

strength.

Masterarbeit Jens Steinwandt

Abstract ii

ABSTRACT

Beamforming and direction of arrival estimation are areas of array signal processing

devoted to extracting information of interest from signals received over the spatial aper-

ture of an antenna array. The need for these techniques is encountered in a broad range

of important engineering applications, including radar, sonar, wireless communications,

radio astronomy and biomedicine, and is still a vital area of research.

In the first part, we focus on the development of widely-linear adaptive beamforming

algorithms designed according to the constrained minimum variance (CMV) principle

for strictly non-circular sources. Specifically, we devise three adaptive algorithms based

on the Krylov subspace, applying the auxiliary vector filtering (AVF), the conjugate

gradient (CG), and the modified conjugate gradient (MCG) algorithms, which avoid

the covariance matrix inversion required by the constraint. The proposed adaptive

algorithms fully exploit the second-order statistics of the strictly non-circular data and

achieve significant performance gains.

The second part is concerned with direction finding algorithms in the beamspace

domain. It is demonstrated that operation in beamspace substantially reduces the

computational complexity while providing increased estimation performance compared

to the element space. Therefore, two Krylov subspace-based algorithms, employing

the AVF and the CG algorithm developed for direction finding, are extended to the

beamspace processing. The development is followed by an extensive performance eval-

uation, highlighting their superior resolution performance for closely-spaced sources at

a low SNR and a small sample size.

In the last part, we introduce a new strategy of incorporating prior knowledge for

direction finding to improve the estimation performance of unknown signal sources.

The novel approach is developed for situations with a limited data record and is based

on an enhanced covariance matrix estimate obtained by linearly combining the sample

covariance matrix and a prior known covariance matrix in an automatic fashion. As

a result, knowledge-aided MUSIC-type and ESPRIT-type direction finding algorithms

are devised and evaluated. Extensive studies to assess the performance illustrate that

the exploitation of prior knowledge is translated into a significantly better estimation

performance.


CONTENTS iii

CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Adaptive beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Direction of arrival estimation . . . . . . . . . . . . . . . . . . . . . . . 31.3 Overview and contributions . . . . . . . . . . . . . . . . . . . . . . . . 41.4 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Data model for direction of arrival estimation . . . . . . . . . . . . . . 72.2 Signal subspace estimation . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 True signal subspace . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Direct data approach . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Covariance approach . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Data model for CMV beamforming . . . . . . . . . . . . . . . . . . . . 11

3. Widely-linear adaptive beamforming using Krylov-based algorithms . . . 133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 LCMV beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Non-circularity and WL-CMV beamforming . . . . . . . . . . . . . . . 16

3.3.1 Non-circularity of signal sources . . . . . . . . . . . . . . . . . . 173.3.2 WL-CMV beamforming assuming strict non-circularity . . . . . 17

3.4 Widely-linear beamforming based on the AVF algorithm . . . . . . . . 193.5 Widely-linear beamforming based on the CG algorithm . . . . . . . . . 213.6 Widely-linear beamforming based on the MCG algorithm . . . . . . . . 243.7 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.7.1 Complexity of LCMV algorithms . . . . . . . . . . . . . . . . . 283.7.2 Complexity of WL-CMV algorithms . . . . . . . . . . . . . . . . 29

3.8 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.8.1 Performance in stationary scenarios . . . . . . . . . . . . . . . . 303.8.2 Performance in non-stationary scenarios . . . . . . . . . . . . . 32


CONTENTS iv

3.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4. Beamspace direction finding using Krylov subspace-based algorithms . . 354.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Beamspace processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Beamspace transformation matrix . . . . . . . . . . . . . . . . . . . . . 39

4.3.1 Discrete Fourier transform (DFT) beamspace . . . . . . . . . . 394.3.2 Discrete prolate spheroidal sequences (DPSS) beamspace . . . . 404.3.3 Comparison of the DFT and DPSS spatial filters . . . . . . . . 41

4.4 Beamspace direction finding based on the CG algorithm . . . . . . . . 414.5 Beamspace direction finding based on the AVF algorithm . . . . . . . . 434.6 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.6.1 Subspace estimation via SVD . . . . . . . . . . . . . . . . . . . 464.6.2 Subspace estimation via EVD . . . . . . . . . . . . . . . . . . . 48

4.7 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.7.1 Comparison of the proposed algorithms in the DFT beamspace . 504.7.2 Comparison of the DFT and DPSS beamspace . . . . . . . . . . 54

4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5. Knowledge-aided direction of arrival estimation . . . . . . . . . . . . . . . 575.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2 Enhanced covariance matrix estimation . . . . . . . . . . . . . . . . . . 59

5.2.1 Computation of the a priori covariance matrix . . . . . . . . . . 595.2.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . 605.2.3 Computation of the optimal weight factors . . . . . . . . . . . . 615.2.4 Estimation of the optimal weight factors . . . . . . . . . . . . . 655.2.5 Improved estimation of the optimal weight factors . . . . . . . . 65

5.3 General knowledge-aided direction of arrival estimation scheme . . . . . 675.3.1 Structure of KA DOA estimation . . . . . . . . . . . . . . . . . 685.3.2 Computational complexity . . . . . . . . . . . . . . . . . . . . . 69

5.4 Knowledge-aided ESPRIT-type direction of arrival estimation . . . . . 705.4.1 KA-ESPRIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.4.2 Unitary KA-ESPRIT . . . . . . . . . . . . . . . . . . . . . . . . 73

5.5 Knowledge-aided MUSIC-type direction of arrival estimation . . . . . . 785.5.1 KA-MUSIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.5.2 KA-Root-MUSIC . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.6 Simulations with accurate a priori knowledge . . . . . . . . . . . . . . 835.6.1 Comparison of LC1 and LC2 using KA-ESPRIT . . . . . . . . . 835.6.2 Comparison of the proposed KA DOA estimation algorithms . . 885.6.3 Closely-spaced signal sources . . . . . . . . . . . . . . . . . . . . 925.6.4 Widely-spaced signal sources . . . . . . . . . . . . . . . . . . . . 925.6.5 Correlated signal sources . . . . . . . . . . . . . . . . . . . . . . 93

5.7 Simulations with inaccurate a priori knowledge . . . . . . . . . . . . . 945.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6. Summary and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 97


CONTENTS v

Appendix A. Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A.1 Proof of equation (3.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A.2 Real-valued source symbols for strict non-circularity . . . . . . . . . . . 100A.3 Proof of equation (3.23) . . . . . . . . . . . . . . . . . . . . . . . . . . 101A.4 Proof of equation (3.43) . . . . . . . . . . . . . . . . . . . . . . . . . . 101A.5 Proof of equation (5.30) . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Symbols and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Theses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Erklärung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111


LIST OF FIGURES vi

LIST OF FIGURES

2.1 Generic model of a ULA . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Block diagram of a narrowband beamformer . . . . . . . . . . . . . . . 11

3.1 Contour lines of complex Gaussian random variables with different ρ . . 17

3.2 Structure of the WL-AVF algorithm . . . . . . . . . . . . . . . . . . . . 19

3.3 Structure of the WL-CG algorithm . . . . . . . . . . . . . . . . . . . . 22

3.4 Structure of the WL-MCG algorithm . . . . . . . . . . . . . . . . . . . 25

3.5 Computational complexity of LCMV and WL-CMV algorithms . . . . . 28

3.6 Output SINR versus the snapshot number in a stationary scenario . . . 31

3.7 Output SINR versus the input SNR in a stationary scenario . . . . . . 31

3.8 Output SINR versus the snapshot number in a non-stationary scenario 32

4.1 Diagram of the beamspace environment . . . . . . . . . . . . . . . . . . 37

4.2 Comparison of the DFT and DPSS spatial filters . . . . . . . . . . . . . 41

4.3 Complexity in terms of arithmetic operations applying the SVD . . . . 48

4.4 Complexity in terms of arithmetic operations applying the EVD . . . . 50

4.5 Probability of resolution versus the SNR in the DFT beamspace . . . . 52

4.6 Probability of resolution versus the snapshots in the DFT beamspace . 52

4.7 RMSE versus the SNR in the DFT beamspace . . . . . . . . . . . . . . 53

4.8 Comparison of the DFT and DPSS beamspace regarding the probability

of resolution versus the SNR . . . . . . . . . . . . . . . . . . . . . . . . 54

4.9 Comparison of the DFT and DPSS beamspace regarding the probability

of resolution versus the snapshot number . . . . . . . . . . . . . . . . . 55

4.10 Comparison of the DFT and DPSS beamspace regarding the RMSE

versus the SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 Block diagram of the KA DOA estimation scheme . . . . . . . . . . . . 68

5.2 Comparison of the LC1 and LC2 regarding the RMSE versus the SNR . 84

5.3 Combination factors of the LC1 and LC2 versus the SNR . . . . . . . . 84

5.4 Comparison of the LC1 and LC2 regarding the RMSE versus the snapshots 85

5.5 Combination factors of the LC1 and LC2 versus the snapshot number . 85


LIST OF FIGURES vii

5.6 RMSE versus the SNR of KA-ESPRIT and forward-backward averaging 87

5.7 RMSE versus the snapshot number of KA-ESPRIT and forward-backward

averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.8 Comparison of KA-ESPRIT-type algorithms . . . . . . . . . . . . . . . 89

5.9 Comparison of Unitary KA-ESPRIT-type algorithms . . . . . . . . . . 89

5.10 Comparison of all the proposed KA algorithms . . . . . . . . . . . . . . 90

5.11 Comparison of KA-MUSIC-type and constrained MUSIC-type algorithms 91

5.12 Comparison of the proposed KA algorithms for closely-spaced sources . 92

5.13 Comparison of the proposed KA algorithms for widely-spaced sources . 93

5.14 Comparison of the proposed KA algorithms for correlated sources . . . 94

5.15 Comparison of the proposed KA algorithms with inaccurate knowledge 95


LIST OF TABLES viii

LIST OF TABLES

3.1 The proposed WL-AVF algorithm . . . . . . . . . . . . . . . . . . . . . 21

3.2 The proposed WL-CG algorithm . . . . . . . . . . . . . . . . . . . . . 24

3.3 The proposed WL-MCG algorithm . . . . . . . . . . . . . . . . . . . . 27

3.4 Computational complexity of LCMV beamforming algorithms . . . . . 29

3.5 Computational complexity of WL-CMV beamforming algorithms . . . . 29

4.1 The proposed DFT BS CG algorithm . . . . . . . . . . . . . . . . . . . 43

4.2 The proposed DFT BS AVF algorithm . . . . . . . . . . . . . . . . . . 45

4.3 Number of additions applying the SVD . . . . . . . . . . . . . . . . . . 47

4.4 Number of multiplications applying the SVD . . . . . . . . . . . . . . . 47

4.5 Number of additions applying the EVD . . . . . . . . . . . . . . . . . . 49

4.6 Number of multiplications applying the EVD . . . . . . . . . . . . . . . 49

5.1 Computational complexity of the proposed versions of the KA scheme . 69

5.2 The proposed KA-ESPRIT algorithm . . . . . . . . . . . . . . . . . . . 72

5.3 The proposed Unitary KA-ESPRIT algorithm . . . . . . . . . . . . . . 77

5.4 The proposed KA-MUSIC algorithm . . . . . . . . . . . . . . . . . . . 80

5.5 The proposed KA-Root-MUSIC algorithm . . . . . . . . . . . . . . . . 82


1. Introduction 1

1. INTRODUCTION

Array signal processing is an active area of research in the broad field of signal process-

ing, and focuses on the problem of estimating signal parameters from data collected

over the spatial aperture of an antenna array, where the sensors are placed at distinct

spatial locations. The given estimation task is usually associated with the extraction

of desired information from impinging signals in the presence of noise and interference.

The antenna array addresses the estimation problem by exploiting the spatial sepa-

ration of the sensor elements to capture the propagating wavefronts, which originate

from energy-radiating sources. Common signal parameters of interest to be estimated

are the signal content itself, the directions of arrival of the signals, and their power. To

obtain this information, the sensor array data is processed using statistical and adap-

tive signal processing techniques. These schemes include parameter estimation and

adaptive filtering applied to array signal processing. The underlying set of principles

and techniques for sensor array signal processing is applicable in many versatile areas

[1, 2], including wireless communications, radar, sonar, biomedicine, seismology and

astronomy.

Among the most relevant topics within array signal processing are beamforming

[1] and direction of arrival estimation [1], which present inevitable challenges when

designing wireless communications systems. As this thesis focuses on the advancement

of these techniques for communications, the main problems to be solved are interference

suppression, source localization, source tracking, complexity reduction and general

performance improvements. For the sake of simplicity, the novel strategies developed

in this work rely on the use of a uniform linear sensor array. An extension to arbitrary

array geometries will be addressed in future work.

1.1 Adaptive beamforming

A ubiquitous task in array signal processing is the extraction of desired information

from propagating signals originating from a certain direction by employing an antenna

array equipped with multiple sensors. The array records the radiating wavefronts and

produces an observation vector at a given time instant, which is processed to obtain the


1. Introduction 2

information of interest. The most straightforward strategy to achieve this objective is

the weighted linear combination of the signals received at multiple spatially separated

sensor elements. By selecting the appropriate weights, signals from a particular direc-

tion can be emphasized while attenuating interfering signals from other directions. This

weighting process is referred to as beamforming or spatial filtering [2, 3]. Generally,

the steering of beams into specific angles by constructing the weights is equivalent to

designing a frequency-selective finite impulse response (FIR) filter for temporal signals.

The weighting applied to the received signal at each sensor element can either be fixed

and data-independent, or continuously adapted to track changes in the system and

to reject interference. The latter approach, termed adaptive beamforming, evidently

provides a better performance and is the focus of this thesis.

The adaptive algorithms used to adjust the weights for each sensor element are

designed by optimizing certain criteria according to the given properties. One of the

most relevant design criteria in practice is the constrained minimum variance (CMV)

approach [4], which does not require the transmission of a known training sequence

and only implies knowledge of the array geometry and the angle of the desired signal.

The CMV optimality criterion minimizes the total beamformer output power while

constraining the array response in the direction of the desired signal to be constant.

Due to its simplicity and effectiveness much effort has been devoted over the past few

decades to devise efficient adaptive algorithms in order to realize a practical beam-

former design [1,3,5]. Among the existing adaptive algorithms, the least mean squares

(LMS) method [5] as a representative of the low-complexity stochastic gradient tech-

niques adopts gradient vectors for the iterative computation of the weights and yields

an acceptable performance in many applications. However, its efficacy strongly de-

pends on the step size and the eigenvalue spread of the covariance matrix, resulting

in an insufficient convergence performance for certain scenarios [3]. An alternative

method is the recursive least squares (RLS) algorithm [5], which is independent of the

eigenvalue spread and thus attains a fast convergence speed. Nevertheless, its main

drawbacks are numerical instability and a relatively high complexity. An attractive

trade-off between the performance and the complexity of the LMS method and the

RLS technique is achieved by the conjugate gradient (CG) algorithm [6, 7] originally

derived to solve a linear system of equations. It forms successive residual vectors within

a number of iterations in order to update the beamforming weight vector, requiring a

lower computational cost compared to the RLS method for few iterations and provid-

ing a faster convergence rate than the LMS approach. A modified version of the CG

(MCG) algorithm derived to obtain further reduction of the complexity while main-

taining the same performance level was proposed in [8]. Another adaptive algorithm


1. Introduction 3

that has received considerable interest is the auxiliary vector filtering (AVF) technique

[9]. Similarly to the CG method, it utilizes an iterative procedure to update the weight

vector by adopting auxiliary vectors to approach the optimal solution. The AVF algo-

rithm was shown to outperform the performance of the RLS method with regard to the

convergence speed and the steady-state level, and to be the state-of-the-art algorithm

for adaptive beamforming [9].

1.2 Direction of arrival estimation

A second problem, which is extensively studied as part of the general field of array

signal processing, is the estimation of the direction of arrival (DOA). The objective

is to determine the angle of arrival of a given spatially propagating signal relative to

the antenna array. To this end, the spatial separation of multiple sensor elements

is exploited to obtain the location of the energy-radiating source. The result of the

estimation procedure is subsequently used for the beamforming design described in the

previous section to steer the beam towards this specific direction, in order to capture

or radiate maximal power. With the field of applications involving DOA estimation

constantly expanding, numerous direction finding techniques have been devised over

the past few decades [1, 10]. The most well-known parameter estimation strategies

discussed here can be classified into three main categories, namely conventional [11],

subspace-based [12–17], and maximum likelihood (ML) methods [18].

The concept of the conventional DOA estimation algorithms relies on the beam-

forming principle. These techniques successively steer the main beam in all possible

look directions and measure the output power [10], which is recorded in the form of a

pseudo spectrum over the angle range. The largest peaks in the pseudo spectrum are

associated with the DOA estimates. The most prominent approach within this class is

Capon’s method [11] based on the CMV criterion [4]. It minimizes the power induced

by interfering signals and noise while keeping the gain towards the look direction fixed.

Although the implementation of the conventional techniques is simple, they suffer from

a lack of angular resolution and require a large number of sensors to achieve a higher

resolution.

The class of subspace-based methods exploits a spectral decomposition of the covari-

ance matrix to achieve high-resolution DOA estimates. Among the most relevant tech-

niques are the multiple signal classification (MUSIC) [12], its extension Root-MUSIC

[13], the estimation of signal parameters via rotational invariance techniques (ESPRIT)

[14], its enhancement Unitary ESPRIT [15] and the recently proposed auxiliary vector

filtering (AVF) [16] and conjugate gradient (CG) [17] algorithms developed for direction


1. Introduction 4

finding. The MUSIC-type and the ESPRIT-type algorithms exploit the eigen-structure

of the covariance matrix, allowing a decomposition of the observation space into a sig-

nal subspace and a complementary noise subspace. Specifically, MUSIC scans over the

possible angle range and makes use of the orthogonality of the subspaces to obtain a

pseudo spectrum with increased resolution. The extension Root-MUSIC for uniform

linear arrays avoids the exhaustive search for peaks by applying a polynomial rooting

technique. The ESPRIT-type algorithms avoid the exhaustive peak search by dividing

the sensor array into two identical subarrays and benefit from the uniform displacement

of the subarrays. The Unitary ESPRIT additionally takes advantage of the fact that

the phase factors representing the displacement lie on the unit circle. The recently

developed AVF and CG algorithms iteratively generate an extended non-eigen-based

signal subspace containing the true signal subspace and the scanning vector itself. The

DOA estimates are determined by the search for the collapse of the extended signal

subspace as the scanning vector belongs to it. Whereas the AVF algorithm adopts aux-

iliary vectors to form the extended signal subspace, the CG method applies residual

vectors and can be considered as an extension of the AVF technique. Both approaches

provide high-resolution estimates at a low signal-to-noise ratio, and a small sample

size.

ML-type methods are based on a parametric approach. They effectively exploit

the underlying data model, resulting in sufficiently high accuracy superior to the con-

ventional and subspace-based methods. However, the efficiency is at the expense of

the computational intensity as a multidimensional search is required. An iterative ap-

proach to limit the computational effort is the alternating projection technique [18],

which transforms the optimization problem into a sequence of one-dimensional opti-

mization problems.

1.3 Overview and contributions

In this work, we deal with the advancement of array signal processing algorithms for the

applications of beamforming and direction finding, which are of significant relevance

in wireless communications.

Chapter 2 introduces the system model for array signal processing, focusing on

the subspace estimation for direction finding and the CMV criterion-based design for

adaptive beamforming.

Chapter 3 is concerned with the first major contribution, which is the develop-

ment of adaptive beamforming methods using Krylov-based algorithms for non-circular


1. Introduction 5

sources. By exploiting the widely-linear property that some of the signal sources are

non-circular, an improved performance can be achieved. Three efficient widely-linear

adaptive algorithms based on the AVF, the CG and the MCG algorithms, which avoid

the computation of the inverse of the covariance matrix are derived according to the

CMV criterion. Furthermore, their performance compared to the linear versions is

investigated and the complexity analyzed.

Chapter 4 deals with the extension of the Krylov subspace-based DOA estimation

algorithms to the operation in beamspace, which is the second major contribution. The

operation in the beamspace domain allows for a significant reduction of the complexity

while substantially increasing the estimation performance. The AVF and CG-based

direction finding techniques are developed for two different beamspace designs, provid-

ing improved performance over their counterparts in element space. Complexity and

performance assessments are also conducted.

Chapter 5 addresses the third contribution, focusing on knowledge-aided DOA

estimation. By exploiting a priori knowledge of some of the DOAs to be estimated,

the estimation accuracy of the unknown signal directions can be significantly enhanced.

A novel way of incorporating the prior knowledge for situations of a limited data

record and a low signal-to-noise ratio is developed. The proposed scheme is applied

to ESPRIT-type and MUSIC-type algorithms and the impact of the knowledge on the

estimation performance is extensively studied.

Chapter 6 concludes the key properties covered in this work, emphasizing the

relevance for the scientific community and discussing possible topics for future work.

1.4 List of publications

Some of the research work outlined in this thesis has already been published. A list of

the conference papers is provided below.

Conference Papers

1. J. Steinwandt, R. C. de Lamare, L. Wang, N. Song, and M. Haardt, “Widely

linear adaptive beamforming algorithm based on the conjugate gradient method,”

International ITG Workshop on Smart Antennas (WSA), (Aachen, Germany),

Feb. 2011.


1. Introduction 6

2. J. Steinwandt, R. C. de Lamare, and M. Haardt, “Beamspace direction finding

based on the conjugate gradient algorithm,” International ITG Workshop on

Smart Antennas (WSA), (Aachen, Germany), Feb. 2011.

3. N. Song, J. Steinwandt, L. Wang, R. C. de Lamare, and M. Haardt, “Non-

data-aided adaptive beamforming algorithm based on the widely linear auxiliary

vector filter,” IEEE International Conference on Acoustics, Speech, and Signal

Processing (ICASSP), (Prague, Czech Republic), May 2011.


2. System Model 7

2. SYSTEM MODEL

This chapter presents the system model of the antenna array output at the receiver,

which provides the basis for the development of array signal processing algorithms.

Although the problems of adaptive beamforming and DOA estimation are different, the

fundamental model used for them is very similar. As stated earlier, the developments

throughout this work are based on the use of a uniform linear array (ULA). Hence, the

introduced system model is only valid for this class of array geometries.

2.1 Data model for direction of arrival estimation

Considering a spherical coordinate system, the DOA of an incident wavefront can be

determined by estimating the azimuth angle θ and the elevation angle φ. For the

sake of simplicity, we limit this model to the one-dimensional case, assuming that the

wavefront impinges on the sensor array in the horizontal plane, i.e. φ = π/2, so that

the azimuth angle θ entirely specifies the DOA. Corresponding to the schematic array

structure in Figure 2.1, additional assumptions to describe the system model are to be

made:

• The propagating wavefronts are assumed to be radiated by point sources.

• The sources are considered to be in the far field and the wavefronts are approxi-mately planar.

x1 x2 xM

∆sin θ1

∆

θ1

s1sd

x3

Fig. 2.1: Generic model of a ULA.


2. System Model 8

• The signals are narrowband signals, so that a time delay corresponds to a phaseshift.

• The sensors are omni-directional, there is no coupling between them and thearray is perfectly calibrated.

Let us suppose that an M -element sensor array receives d (d < M) propagating wave-

fronts with the distinct DOAs θ = [θ1, . . . , θd]T relative to the broad side of the array.

The ith of N available data snapshots of the (M × 1)-dimensional array output vectorcan be modeled as

x(i) =[a(θ1) · · · a(θd)

]

s1(i)...

sd(i)

+ n(i) = A(θ)s(i) + n(i), i = 1, . . . , N, (2.1)

where A(θ) ∈ CM×d is the array steering matrix, whose columns are functions of theunknown DOAs. The zero-mean vector s(i) ∈ Cd×1 represents the impinging complexwaveforms of the d signals and n(i) ∈ CM×1 is the additive vector of white circularlysymmetric complex Gaussian noise with zero mean and variance σ2n.

Considering a ULA equipped with equally spaced sensors as shown in Figure 2.1,

the (M × 1) steering vectors a(θn) corresponding to the nth source, n = 1, . . . , d, canbe expressed as

a(θn) =[1 ej2π

∆λc

sin θn · · · ej2π(M−1) ∆λc sin θn]T, (2.2)

where ∆ denotes the distance of adjacent sensors of the ULA and λc is the signal

wavelength. In order to avoid spatial aliasing, ∆ is restricted to ∆ ≤ λc/2 in whichcase the DOAs are limited to the interval −90◦ < θn < 90◦. Defining the spatialfrequencies µn ∈ [−π, π] as a result of the geometry in Figure 2.1 as

µn = 2π∆

λcsin θn, (2.3)

the model stated in (2.1) can be written as

x(i) =

1 1 · · · 1ejµ1 ejµ2 · · · ejµd

......

...

ej(M−1)µ1 ej(M−1)µ2 · · · ej(M−1)µd

s(i) + n(i), (2.4)


2. System Model 9

where the array steering matrix A(θ) exhibits the Vandermonde structure. Given the

entire data of the N snapshots, (2.4) can also be expressed compactly as

X =[x1 · · · xN

]= A(θ)S + N ∈ CM×N . (2.5)

2.2 Signal subspace estimation

The DOA estimation techniques considered in this work belong to the class of subspace-

based algorithms. Therefore, a brief review of the fundamental properties is covered in

this section.

2.2.1 True signal subspace

Due to the fact that the source data s(i) and the noise vector n(i) are modeled as

uncorrelated random variables, the covariance matrix of the noise-corrupted zero-mean

array output x(i) is calculated by

R = E{x(i)xH(i)

}= A(θ)RssA

H(θ) + σ2nIM ∈ CM×M , (2.6)

where Rss = E{s(i)sH(i)} is the signal covariance matrix, which is diagonal if thesources are uncorrelated and nondiagonal for partially correlated sources. The noise

covariance matrix E{n(i)nH(i)} = σ2nIM , where IM denotes the M × M identitymatrix.

The eigenvalue decomposition (EVD) of the covariance matrix R can be expressed

as

R =[Us Un

]Λd 0

0 0

+ σ2nIM

U

Hs

UHn ,

(2.7)

where the diagonal matrix Λd = diag{λn}dn=1 consists of the nonzero eigenvalues,and Us and Un represent the orthogonal eigenvectors corresponding to the d largest

eigenvalues and the M − d smallest eigenvalues, respectively. The matrix Us is oftenreferred to as signal subspace S and its orthogonal complement Un is termed noisesubspace. Without additive noise, i.e., σ2n = 0, it is evident from

R = A(θ)RssAH(θ) = UsΛdU

Hs (2.8)

that A(θ) and Us span the same signal subspace S of dimension d, i.e.,

S = span{A(θ)} = span{Us}. (2.9)


2. System Model 10

2.2.2 Direct data approach

In the direct data approach, a singular value decomposition (SVD) of the array output

matrix X is performed to estimate the signal and the noise subspace. Assuming that

the number of sources d, i.e., the rank, is known, X can be written as

X =[Ûs Ûn

]Σ̂s 0

0 Σ̂n

V̂

Hs

V̂ Hn

(2.10)

= ÛsΣ̂sV̂Hs + ÛnΣ̂nV̂

Hn , (2.11)

where the diagonal matrices Σ̂s and Σ̂n contain the d largest singular values and the

M − d smallest singular values, respectively. The estimated signal subspace Ûs ∈CM×d comprises the singular vectors corresponding to Σ̂s and the noise subspace Ûn ∈

CM×(M−d) is associated with Σ̂n. If only the signal subspace is to be estimated a rank-d

approximation of the SVD can be applied.

2.2.3 Covariance approach

In practical applications, the true covariance matrix R is unknown and needs to be

estimated. A widely used estimator for R is a sample-averaged approach, termed

sample covariance matrix, which is given by

R̂ =1

N

N∑

i=1

x(i)xH(i) =1

NXXH ∈ CM×M . (2.12)

Again, assuming the rank d is known, the EVD of the sample covariance matrix R̂

yields

R̃ =[Ûs Ûn

]Λ̂s 0

0 Λ̂n

Û

Hs

ÛHn

(2.13)

= ÛsΛ̂sÛHs + ÛnΛ̂nÛ

Hn , (2.14)

where the diagonal matrices Λ̂s and Λ̂n contain the d largest eigenvalues and the

M − d smallest eigenvalues, respectively. The estimated signal subspace Ûs ∈ CM×dis composed of the eigenvectors corresponding to Σ̂s and the noise subspace Ûn ∈CM×(M−d) complies with Σ̂n. If only the signal subspace is to be estimated a rank-d

approximation of the EVD can be applied. Both the direct data approach and the

covariance approach are theoretically equivalent.


2. System Model 11

2.3 Data model for CMV beamforming

In order to describe adaptive beamforming techniques according to the constrained

minimum variance (CMV) principle, the same assumptions on the impinging wavefronts

as in the previous section are presumed. However, for the system model, we do not

restrict the sensor array to be a ULA as this property is not required by the adaptive

algorithms. Furthermore, we use a slightly different notation to distinguish between

the problems of DOA estimation and adaptive beamforming.

Assume that d narrowband signals from sources in the far field are impinging

on an arbitrarily formed array of M (M ≥ d) sensor elements with the DOAs θ =[θ1, . . . , θd]

T . The ith of N available data snapshots of the (M × 1)-dimensional arrayoutput vector can be modeled as

r(i) = A(θ)P 1/2s(i) + n(i), i = 1, . . . , N, (2.15)

where the zero-mean vector s(i) = [s1(i), . . . , sd(i)]T ∈ Cd×1 represents the complex

signal waveforms, P = diag{σ21, . . . , σ2d} contains the signal powers on its diagonal,n(i) ∈ CM×1 is the vector of white circularly symmetric complex Gaussian noise withzero mean and variance σ2n, and the matrix A(θ) = [a(θ1), . . . ,a(θd)] ∈ CM×d consistsof the array steering vectors corresponding to the d sources.

Corresponding to Figure 2.2, the output of the adaptive beamformer is generated

by appropriately weighting the sum of the signals at each sensor element. Then, the

output signal is sent back to the adaptive processor to update the weights subject

to the optimization of a cost function. The weighting process can be mathematically

θn

x1(i)

x2(i)

xM (i)

1

2

M

Sources

Sensors1

2

d

∑∑∑

w∗1

w∗2

w∗M

y(i)

Adaptive

Algorithm

Fig. 2.2: Block diagram of a narrowband beamformer.


2. System Model 12

expressed as

y(i) = wHr(i), (2.16)

where w = [w1, . . . , wM ]T ∈ CM×1 is the complex beamforming weight vector to be

designed. After estimating the DOAs in the environment and selecting a particular

source of interest with the signal direction θ1, the beamformer output in (2.16) can be

expanded as

y(i) =√σ21w

Ha(θ1)s1(i) + wH

(K∑

k=2

√σ2ka(θk)sk(i) + n(i)

)(2.17)

=√σ21w

Ha(θ1)s1(i) + wHu(i), (2.18)

where u(i) is the vector containing the interferers and the additive noise. The objective

is to keep the gain in the direction of the desired user fixed, i.e., wHa(θ1) = γ and the

gain of the interferers small wHu(i) ≈ 0. This can be achieved by CMV beamforming,which formulates the objective as the constrained optimization problem

wopt = argminw

E

{|y(i)|2

}= E

{|wHr(i)|2

}= wHRw

subject to wHa(θ1) = γ,(2.19)

where the covariance matrix R is estimated by the sample covariance matrix R̂ defined

in (2.12). The goal of (2.19) is to minimize the array output variance while maintaining

the response in the direction of the SOI.

Applying the method of Lagrange multipliers, the optimal solution for the mini-

mization problem is

wopt =γR−1a(θ1)

aH(θ1)R−1a(θ1). (2.20)

Setting the constant γ = 1 yields the so-called minimum variance distortionless re-

sponse (MVDR). Note that the CMV criterion requires the knowledge of the DOA θ1

of the desired user and the array geometry to form the steering vector a(θ1). Further

details and a classification into linear CMV (LCMV) and widely-linear CMV (WL-

CMV) will be discussed in the next chapter.


3. Widely-linear adaptive beamforming using Krylov-based algorithms 13

3. WIDELY-LINEAR ADAPTIVE

BEAMFORMING USING

KRYLOV-BASED ALGORITHMS

In this chapter, we develop three widely-linear constrained adaptive beamforming al-

gorithms based on the Krylov subspace, which are designed according to the minimum

variance principle. The proposed algorithms apply either the auxiliary vector filter-

ing (AVF) algorithm, the conjugate gradient (CG) algorithm or the modified conju-

gate gradient (MCG) algorithm to iteratively update the beamforming weight vector.

By exploiting the non-circularity of impinging signals via a widely-linear processing

technique, the output performance of these adaptive algorithms can be significantly

improved.

3.1 Introduction

Adaptive beamforming is an essential approach for estimating the properties of a signal

of interest at the sensor array output by weighting and combining the received signals

to filter the desired user and to reject interference. The key advantage of the adaptive

architecture is its ability to automatically adjust its weights to suit differences in the

observed scenario. This includes tracking of moving interferers, or adaptation to leaving

or entering signal sources in the system. Given its vast importance in multiple fields

such as radar, sonar, biomedicine and wireless communications, adaptive beamforming

has received considerable attention in the last few decades and numerous algorithms

have been reported [5]. One of the most popular criteria for designing a beamformer is

the linearly constrained minimum variance (LCMV) concept [4], which minimizes the

variance at the array output while maintaining a desired response in the direction of

the source of interest (SOI). This design criterion only requires the knowledge of the

steering vector of the desired signal.

In most existing estimation and filtering algorithms, it is often assumed that the re-

ceived vector r(i) is second-order circular as for quadrature phase-shift keying (QPSK)-



modulated signals, where the conventional covariance matrix

R = E{r(i)rH(i)

}∈ CM×M (3.1)

fully describes the second-order statistics. However, in many modern telecommunica-

tions and satellite systems based on binary phase-shift keying (BPSK)-modulated or

amplitude-shift keying (ASK)-modulated signals, the observation signal r(i) features

second-order non-circularity, i.e., the complementary covariance matrix

C = E{r(i)rT (i)

}6= 0 ∈ CM×M (3.2)

and thus needs to be considered to optimally characterize the statistical properties of

the signals [19]. It was recently shown that fully exploiting the second-order statis-

tics, referred to as widely-linear (WL) processing, substantially improves the output

performance compared to that in the linear processing [19–21]. Consequently, several

widely-linear adaptive beamforming algorithms comprising the WL minimum variance

distortionless response (WL-MVDR) algorithm [21] and the WL recursive least squares

(WL-RLS) algorithm [22] have been devised. These approaches are either based on the

assumption of weak non-circularity [21] or strict non-circularity [22] of the signals. A

slight drawback associated with the strategy of widely-linear processing is the increased

complexity due to the extended dimensions as both the originally observed data vector

r(i) and its complex conjugate r∗(i) need to be processed. As the filtered data ordi-

narily occupies a large sample size and also, largely equipped sensor arrays are utilized

in practice, the widely-linear processing scheme further slows down the convergence

speed of the adaptive algorithms.

Among the well-established linearly constrained adaptive algorithms are the least

mean squares (LMS) approach [5], which requires a low computational complexity but

exhibits a slow convergence, and the recursive least squares (RLS) algorithm [5] that

converges fast but suffers from a high computational burden. Two more recently de-

veloped adaptive beamforming algorithms, namely the auxiliary vector filtering (AVF)

algorithm [9] and the conjugate gradient (CG) algorithm [8] [23], rely on the Krylov

subspace and iteratively update the filter weights in a stochastic gradient fashion. The

AVF algorithm is known to yield a high convergence speed and to outperform the

LMS and the RLS techniques. The iterative method based on the CG algorithm can

provide an attractive trade-off between the convergence performance and the compu-

tational complexity. Similarly to the AVF algorithm, it avoids the computation of

R−1 required by the constraint, saving significant computational resources. However,

it does not attain the performance level of the RLS algorithm. Despite the benefits



of these Krylov-based methods, they require several iterations per data snapshot in

order to converge. Thus, a modified version of the CG algorithm employing only one

iteration per data update was developed in [23]. By making use of a degenerated

scheme, meaning that its residual vectors are not requested to be orthogonal to the

spanned Krylov subspace, the computational cost is further reduced while maintaining

the performance.

Motivated by the convergence speed-enhancing and complexity-reducing perfor-

mance of the Krylov-based adaptive algorithms, this chapter is concerned with the

development of these algorithms combined with the strategy of widely-linear process-

ing to overcome the issues of the increased complexity and the convergence speed

limitations. The resulting novel algorithms devised for strictly non-circular sources are

referred to as the WL-AVF, the WL-CG and the WL-MCG beamforming methods and

are designed according to the widely LCMV (WL-CMV) criterion. Requiring only the

a priori knowledge of the signal direction of the desired user, the proposed techniques

fully exploit the second-order statistics of the strictly non-circular signals in the system

and, at the same time, ensure the benefits of the Krylov-based algorithms. In connec-

tion with the development of the WL-AVF, the WL-CG, and the WL-MCG methods,

an analysis of their computational complexity is performed and their convergence ca-

pabilities are studied. For comparison purposes, their performance is contrasted with

the previously developed WL-LMS and the WL-RLS algorithms.

3.2 LCMV beamforming

We start by shortly reviewing the LCMV concept for adaptive beamforming. Adopting

the definition of the received vector r(i) stated in Section 2.3, the output of an adaptive

narrowband beamformer is given by

y(i) = wHr(i), (3.3)

where

w = [w1, . . . , wM ]T ∈ CM×1 (3.4)

is the beamforming vector to be designed, which contains the complex weights that

multiply the signals at each sensor element. The adaptation of the weight vector

according to the LCMV optimality criterion is defined as the minimization of the

total output variance while simultaneously keeping the gain of the array into a desired

signal direction θ1 fixed. This formulation is equivalent to the reduction of the output

variance by suppressing the interference. Mathematically, the optimal weights wopt are



computed by solving the constrained optimization problem

wopt = argminw

E

{|y(i)|2

}= E

{|wHr(i)|2

}= wHRw

subject to wHa(θ1) = γ,(3.5)

where R is the M × M covariance matrix defined in (3.1), a(θ1) is the array steeringvector of the desired user with the DOA θ1, and γ is an arbitrary constant. Solving

the minimization problem by using the method of Lagrange multipliers, the optimal

solution for the calculation of the weights is obtained by

wopt =γR−1a(θ1)

aH(θ1)R−1a(θ1). (3.6)

Setting the constant γ = 1 yields the so-called MVDR beamformer, meaning that

the desired signal is undistorted due to the gain of 1 and the output signal exhibits

minimum variance. Note that the LCMV criterion requires the knowledge of the DOA

of the SOI and the exact array geometry to form the steering vector a(θ1).

In order to assess the effectiveness of an adaptive beamformer, the signal-to-interference-

plus-noise ratio (SINR) for the particular signal direction θ1 at each snapshot i is

analyzed. The expression of the SINR is given by

SINR(i) =σ21|wH(i)a(θ1)|2E {|wH(i)v(i)|2} =

σ21|wH(i)a(θ1)|2wH(i)Ruw(i)

, (3.7)

where σ21 is the power of the SOI and Ru is the M × M covariance matrix of theinterference plus noise. The maximum output SINR according to the LCMV criterion

can be derived as

SINRLCMVmax =σ21a

H(θ1)R−1a(θ1)

1 − σ21aH(θ1)R−1a(θ1). (3.8)

The proof of equation (3.8) was moved to the Appendix A.1.

3.3 Non-circularity and WL-CMV beamforming

As the objective of this chapter is to develop WL adaptive beamforming algorithms for

non-circular sources, we first review the properties of non-circularity and then make

use of the resulting characteristics to design the WL-CMV optimization criterion to

generate the beamforming weight vectors.



−2 −1 0 1 2−2

−1

0

1

2

(a) ρ = 0

−2 −1 0 1 2−2

−1

0

1

2

(b) ρ = 0.6 exp(jπ/2)

−2 −1 0 1 2−2

−1

0

1

2

(c) ρ = 1 exp(jπ/2)

Fig. 3.1: Contour lines of complex Gaussian random variables with different ρ.

3.3.1 Non-circularity of signal sources

A signal is termed non-circular if the real and the imaginary part of its source symbols

are dependent, such that the lines of equal probability in the joint density function

are not circles anymore [24, 25]. Consequently, a signal is referred to as circular if the

opposite fact holds. For mathematical purposes, the non-circularity coefficient

ρ =E {s2(i)}E {|s(i)|2} = |ρ|e

jψ (3.9)

is defined to provide a measure for the deviation from the circular symmetry. Thus,

the non-circularity coefficient is ρ = 0 if the signal is circular. Assuming a signal in

the system to be non-circular is not a distinct condition. Furthermore, it is essential to

distinguish between weak non-circularity and strict non-circularity. The requirement

for a weakly non-circular signal is 0 < |ρ| < 1 and a strictly non-circular signal impliesa non-circularity coefficient of |ρ| = 1. The contour lines of constant probability densityfor the three cases depending on ρ are depicted in Figure 3.1. Note that the lines for

non-circular sources can have an arbitrary orientation ψ/2, which is determined by the

angle of (3.9) [24]. In the following development of the WL Krylov-based beamforming

algorithms, we assume the property of strict non-circularity of all the signals in the

system.

3.3.2 WL-CMV beamforming assuming strict non-circularity

In order to exploit the strict non-circularity of signals impinging on the sensor array, the

second-order statistics fully described by the covariance matrix R = E{r(i)rH(i)} andthe complementary covariance matrix C = E{r(i)rT (i)} need to be taken into account.Thus, to make use of the additional information contained in C, the received vector r(i)



and its complex conjugate r∗(i) are processed by applying a bijective transformation

T such thatr(i)

T−→ r̃(i) : r̃(i) = [rT (i), rH(i)]T ∈ C2M×1, (3.10)

where r̃(i) is the augmented received vector. It is utilized by the designed beamformer

to design the augmented weight vector w̃ ∈ C2M×1 and to generate the output y(i) =w̃H r̃(i), which is termed widely-linear beamforming. In what follows, all the widely-

linear quantities are denoted by an over tilde. Expanding the augmented vector after

the bijective transformation, and taking into account the entire data block, yields

X̃ =

XX∗

=

A(θ)SA∗(θ)S∗

+

NN ∗

∈ C2M×N , (3.11)

where we used X to denote the array output matrix for beamforming to avoid confu-

sions with the covariance matrix R. As a result of the strict non-circularity assumption

with |ρ| = 1, the source symbols in S are real-valued and (3.11) can be rewritten as

X̃ =

A(θ)A∗(θ)

S +

NN ∗

. (3.12)

A proof of the implication of real-valued symbols when presuming a non-circularity

coefficient of |ρ| = 1 is shown in Appendix A.2.

Applying the bijective transformation T to the LCMV optimization problem in(3.5), we obtain a new optimization problem referred to as WL-CMV design criterion,

which is formulated as

w̃opt = argminw̃

E

{|y(i)|2

}= E

{|w̃H r̃(i)|2

}= w̃HR̃w̃

subject to w̃Hã(θ1) = γ,(3.13)

where ã(θ1) = T {a(θ1)} is the augmented steering vector of the SOI and R̃ =E{r̃(i)r̃H(i)} ∈ C2M×2M is the augmented covariance matrix of the structure

R̃ =

R C

C∗ R∗

, (3.14)

where the covariance matrix R and the complementary covariance matrix C appear in

conjugate pairs. Solving the widely-linear minimization problem using the Lagrange



multiplier yields the optimal solution given by

w̃opt =γR̃−1ã(θ1)

ãH(θ1)R̃−1ã(θ1). (3.15)

The performance of a widely-linear adaptive beamformer is evaluated through the

WL SINR, which is expressed similarly to (3.7) as

SĨNR(i) =σ21|w̃H(i)ã(θ1)|2w̃H(i)R̃uw̃(i)

, (3.16)

where R̃u is the augmented covariance matrices of the interference plus noise. The

maximum output SINR in the WL case is given by

SINRWL-CMVmax =σ21ã

H(θ1)R̃−1ã(θ1)

1 − σ21ãH(θ1)R̃−1ã(θ1). (3.17)

3.4 Widely-linear beamforming based on the AVF algorithm

In this part, the widely-linear adaptive beamforming algorithm adopting the AVF

technique, termed WL-AVF algorithm, is developed. It utilizes the auxiliary vectors

provided by the AVF algorithm to iteratively update the widely-linear weight vector

w̃. After introducing the general structure of the WL-AVF-based beamforming design,

the algorithm is derived in detail according to the previous work in [9].

Structure of the WL-AVF algorithm for beamforming

The general framework of the proposed beamformer based on the WL-AVF algorithm

is depicted in Figure 3.2. First, the augmented received vector r̃(i) is obtained by

applying a bijective transformation T to the received input vector r(i). The augmentedvector is then processed by the filter, whose complex weights are adjusted via the WL-

FilterDesign

WL-AVFAlgorithm

T {·}r(i) y(i)r̃(i)

wK(i)

Fig. 3.2: Structure of the WL-AVF algorithm.



AVF algorithm to generate the beamformer output y(i). The final weight wK(i) per

data update is computed after K iterations of the WL-AVF algorithm. In the following

derivation, we drop the time instant i to increase the readability.

Proposed WL-AVF algorithm

The development of the WL-AVF algorithm starts with the initialization of the aug-

mented weight vector w̃, which is chosen to be equal to the conventional matched filter

solution for a desired response γ in the direction θ1, given by

w̃0 = γ∗ ã(θ1)

‖ã(θ1)‖2. (3.18)

Then, the weight vectors are iteratively updated in a stochastic gradient manner by

subtracting scaled auxiliary vectors g̃ that are constructed to be orthogonal to ã(θ1).

The successively computed augmented weight vectors are obtained by

w̃k = w̃k−1 − µkg̃k, k = 1, . . . , K, (3.19)

where µk is the step size for the kth iteration. The augmented auxiliary vectors g̃k are

designed to maximize the magnitude of the cross-correlation between g̃Hk r̃ and w̃Hk−1r̃,

subject to the orthonormality constraint with respect to ã(θ1) [9], which is formulated

by the optimization problem

g̃k = argmaxg̃k

∣∣∣E{w̃Hk−1r̃[g̃

Hk r̃]

H} ∣∣∣

= argmaxg̃k

∣∣∣E{w̃Hk−1R̃g̃k

} ∣∣∣

s.t. g̃Hk ã(θ1) = 0, ‖g̃k‖ = 1.

(3.20)

According to [9], the solution to this problem is expressed by

g̃k =

(I − ã(θ1)ãH(θ1)

‖ã(θ1)‖2

)R̃w̃k−1

∥∥∥(I − ã(θ1)ãH(θ1)

‖ã(θ1)‖2

)R̃w̃k−1

∥∥∥(3.21)

Eventually, the value for the step size µk is determined by minimizing the variance at

the output, i.e.,

µk = minµk

E

{|w̃Hk r̃|2

}(3.22)

with the solution

µk =g̃Hk R̃w̃k−1

g̃Hk R̃g̃k. (3.23)



Tab. 3.1: The proposed WL-AVF algorithm

Initialization: w̃0 = γ∗ ã(θ1)

‖ã(θ1)‖2

for each snapshot i = 1, . . . , N

R̃dl(i) = λR̃dl(i − 1) + r̃(i)r̃H(i) + ξI2Mfor iteration k = 1, . . . , K

g̃k =

(I2M −

ã(θ1)ãH (θ1)

‖ã(θ1)‖2

)R̃dlw̃k−1∥∥∥

(I2M −

ã(θ1)ãH (θ1)

‖ã(θ1)‖2

)R̃dlw̃k−1

∥∥∥If ‖g̃k − g̃k−1‖ → 0, EXIT.µk =

g̃Hk

R̃dlw̃k−1

g̃Hk

R̃g̃k

w̃k = w̃k−1 − µkg̃kend

end

For the proof of (3.23), the reader is referred to Appendix A.3.

The proposed beamforming technique based on the WL-AVF algorithm is summa-

rized in Table 3.1, where the time instant i is included again to emphasize the adaptive

approach of obtaining the augmented weight vector for each snapshot. As the auxil-

iary vectors approach zero when k increases, the iterative procedure is terminated if

‖g̃k − g̃k−1‖ → 0.

Applying the developed WL-AVF beamforming algorithm to non-stationary sys-

tems, the covariance matrix R̃ is estimated by its recursive form using the exponentially

weighted estimate:

R̃(i) = λR̃(i− 1) + r̃(i)r̃H(i), (3.24)

where λ is the forgetting factor, which is close to but less than 1. The first initialization

of R̃(0) = δI2M , where a common value for δ = σ2n. Furthermore, the diagonal loading

technique [26] is adopted to achieve more accurate estimates of R̃ for a small number

of available snapshots, i.e.,

R̃dl(i) = R̃(i) + ξI2M , (3.25)

where the diagonal loading factor ξ is typically chosen to be ξ = 10·σ2n according to [27].Other approaches to improve the estimates in short data records include reduced-rank

algorithms as in [28].

3.5 Widely-linear beamforming based on the CG algorithm

This part of the chapter deals with the derivation of the WL-CG algorithm for adaptive

beamforming in the case of strictly non-circular signals. The WL-CG algorithm forms



FilterDesign w̃(i)

WL-CGAlgorithm

T {·}r(i) y(i)r̃(i)

ṽK(i)

Fig. 3.3: Structure of the WL-CG algorithm.

orthogonal residual vectors, which are employed to iteratively update the widely-linear

weight vector w̃(i). Whereas the general structure is described first, the second part is

concerned with the development of the WL-CG method as an extension of the previous

work in [8, 23].

Structure of the WL-CG algorithm for beamforming

The block diagram of the WL-CG-based beamforming procedure depicted in Figure 3.3

exhibits a similar structure as for the WL-AVF algorithm. The bijective transformation

T applied to the original received vector r(i) yields the augmented received vectorr̃(i), which is processed by the filter to provide the output y(i). However, contrary

to the WL-AVF procedure, the WL-CG method iteratively updates a new defined

augmented vector ṽ(i) to avoid the computation of R̃−1 that is passed onto the filter

after performing K iterations of the WL-CG algorithm. Thus, the final beamforming

weight vector w̃(i) is computed in the filter design. Again, we drop the time instant i

in the following development.

The proposed WL-CG algorithm

In order to derive the WL-CG beamforming algorithm for non-circular sources, the

WL-CG method is applied to minimize the cost function [29]

min J(ṽ) = ṽHR̃ṽ − 2Re{ãH(θ1)ṽ

}, (3.26)

where ṽ ∈ C2M×1 is a newly defined augmented CG weight vector, which is differentfrom the beamforming weight vector w̃ to be designed. The CG method was originally

developed to solve the optimization problem in (3.26) [6]. However, by taking the

gradient of (3.26) with respect to the CG weight vector ṽ and equating it to zero,

it can be shown that minimizing the optimization problem in (3.26) is equivalent to



solving the system of linear equations [23]

R̃ṽ = ã(θ1) ∈ C2M×1. (3.27)

The WL-CG algorithm solves (3.27) by approaching the optimal weight solution step

by step via a line search along successive directions, which are sequentially determined

at each iteration [6]. The solution for the WL-CG optimization problem (3.26) is

obtained as

ṽ = R̃−1ã(θ1). (3.28)

As the expression R̃−1ã(θ1) also occurs in the optimal WL-CMV solution in (3.15),

the WL-CMV beamformer can thus be iteratively computed without the calculation

of R̃−1 by updating the new augmented CG weight vector ṽ as

ṽk = ṽk−1 + αkp̃k, k = 1, . . . , K, (3.29)

where p̃k is the direction vector, αk is the step size value to control its weight and k

is the iteration number. The directions p̃k are orthogonal with respect to R̃, termed

R̃-orthogonality, i.e.,

p̃Hk R̃p̃l = 0 (3.30)

for all k 6= l. The following direction was determined in the previous iteration and isobtained by

p̃k+1 = g̃k + βkp̃k, (3.31)

where g̃k is the residual vector defined in [6] that is given by

g̃k = ã(θ1) − R̃ṽk. (3.32)

The step size αk is chosen to minimize the cost function in the determined direction

and βk ensures the R̃-orthogonality of the direction vectors p̃k in (3.30). According to

[6], both parameters are calculated by

αk =g̃Hk−1p̃k

p̃Hk R̃p̃kand βk =

g̃Hk g̃k

g̃Hk−1g̃k−1. (3.33)

After K iterations of the WL-CG algorithm the augmented complex weight vector w̃

is formed as

w̃ =γṽK

ãH(θ1)ṽK. (3.34)

A summary of the proposed WL-CG-based beamforming technique is given in Table



Tab. 3.2: The proposed WL-CG algorithm

Initialization: ṽ0(1) = 0, R̃dl(0) = δI2Mfor each snapshot i = 1, . . . , N

R̃dl(i) = λR̃dl(i − 1) + x̃(i)x̃H(i) + ξI2M ,p̃1(i) = g̃0(i) = ã(θ1) − R̃dl(i)ṽ0(i)for iteration k = 1, . . . , K

d̃k(i) = R̃dl(i)p̃k(i)

αk(i) = [p̃Hk (i)d̃k(i)]

−1g̃Hk−1(i)p̃k(i)ṽk(i) = ṽk−1(i) + αk(i)p̃k(i)

g̃k(i) = g̃k−1(i) − αk(i)d̃k(i)βk(i) = [g̃

Hk−1(i)g̃k−1(i)]

−1g̃Hk (i)g̃k(i)p̃k+1(i) = g̃k(i) + βk(i)p̃k(i)

end

ṽ0(i + 1) = ṽK(i)w̃(i) = [ãH(θ1)ṽK(i)]

−1γṽK(i)end

3.2, where again the included time instant i indicates the adaptive way to obtain the

weight solution for each data snapshot.

As already outlined in the previous section, the covariance matrix R̃ is estimated by

its recursive form using the exponentially weighted estimate and incorporating diagonal

loading:

R̃dl(i) = λR̃dl(i− 1) + r̃(i)r̃H(i) + ξI2M , (3.35)

where the diagonal loading factor ξ is again chosen to be ξ = 10 · σ2n [27].

3.6 Widely-linear beamforming based on the MCG algorithm

In this section, a modified version of the WL-CG method, termed WL-MCG algo-

rithm, is developed for widely-linear adaptive beamforming. Similarly to the WL-CG

algorithm, it applies residual vectors to update the beamforming weight vector in an

iterative fashion. However, it only requires one iteration of the CG algorithm per sam-

ple update. Again, the first part deals with the structure of the MCG method and in

the second part, the detailed algorithm is derived according to [8].

Structure of the WL-MCG algorithm for beamforming

The general structure of the WL-MCG algorithm for adaptive beamforming is shown as

a block diagram in Figure 3.4. The augmented received vector r̃(i) obtained after the

bijective transformation is processed by the WL-MCG algorithm to provide the weight



FilterDesign

WL-MCGAlgorithm

T {·}r(i) y(i)r̃(i)

w̃(i)

Fig. 3.4: Structure of the WL-MCG algorithm.

vector w̃(i) for the filter. The key advantage is the property that, unlike the WL-CG

developed in the previous section, the MCG method requires only one iteration per

data snapshot, saving computational resources. Furthermore it also adopts the new

defined augmented vector ṽ(i) to avoid the costly computation of R̃−1.

The proposed WL-MCG algorithm

Corresponding to the WL-CG beamforming method, the WL-MCG algorithm is an-

other way of solving the system of equations stated in (3.27) in the previous section.

The covariance matrix is estimated by using the diagonal-loading enhanced exponen-

tially weighted data window

ˆ̃Rdl(i) = λ

ˆ̃Rdl(i− 1) + r̃(i)r̃H(i) + ξI2M , (3.36)

where λ is the forgetting factor and ξ the diagonal loading factor. According to [1],

the estimated covariance matrix can be approximated for a large sample size as

ˆ̃Rdl(i) ≈

1

1 − λ(R̃ + ξI2M). (3.37)

Now, we define the new augmented weight vector ṽ(i) to avoid the computationally

expensive computation of ˆ̃R−1

dl (i) at each snapshot as

ṽ(i) = ˆ̃R−1

dl (i)ã(θ1). (3.38)

Inserting equation (3.37) into (3.38), we obtain

ṽ(i) =(

1

1 − λ(R̃ + ξI2M))−1

ã(θ1) (3.39)

for a sufficiently large number of snapshots.

Similarly to the WL-CG algorithm, the WL-MCG method iteratively computes the



solution of (3.38) by updating the new defined augmented vector ṽ(i) as

ṽ(i) = ṽ(i− 1) + α(i)p̃(i), (3.40)

where the direction vector p̃(i) was determined at the previous snapshot by

p̃(i+ 1) = g̃(i) + β(i)p̃(i) (3.41)

and the residual vector g̃(i) is given by

g̃(i) = ã(θ1) − ˆ̃R(i)ṽ(i). (3.42)

In order to realize sample-by-sample processing, a recursive expression for the residual

vector g̃(i) can be derived by taking into account the equations (3.42), (3.36), and

(3.40), leading to

g̃(i) = (1 − λ)ã(θ1) + λg̃(i− 1) −(r̃(i)r̃H(i) + ξI2M

)ṽ(i− 1) − α(i) ˆ̃R(i)p̃(i). (3.43)

The proof of (3.43) is shown in Appendix A.4.

The basic idea of the WL-MCG is to employ a degenerated scheme [8], meaning

that the obtained residual vector g̃(i) is not required to be orthogonal to the subspace

spanned by the previously determined direction vectors S = span{p̃(1), p̃(2), . . . , p̃(i)},i.e., p̃H(i)g̃(l) 6= 0 for l < i. Under this condition the step size α(i) has to satisfy theconvergence bound defined according to [8, 23] as

0 ≤ E{p̃H(i)g̃(i)

}≤ 0.5 E

{p̃H(i)g̃(i− 1)

}. (3.44)

To this end, premultiplying equation (3.43) by p̃H(i), taking the expectation of both

sides and considering (3.39) yields

E

{p̃H(i)g̃(i)

}≈ λE

{p̃H(i)g̃(i− 1)

}− E {α(i)}E

{p̃H(i) ˆ̃R(i)p̃(i)

}, (3.45)

where the convergence of the adaptive algorithm is assumed, such that the approxima-

tion in (3.39) holds, and therefore

(1 − λ)ã(θ1) −[E

{r̃(i)r̃H(i)

}+ ξI2M

]ṽ(i− 1) = 0. (3.46)

Inserting the obtained expression (3.45) into the convergence bound in (3.44) and



Tab. 3.3: The proposed WL-MCG algorithm

Initialization: ṽ(0) = 0, R̃dl(0) = δI2M , p̃(1) = g̃(0) = ã(θ1)for each snapshot i = 1, . . . , N

R̃dl(i) = λR̃dl(i − 1) + x̃(i)x̃H(i) + ξI2M ,d̃(i) = R̃dl(i)p̃(i)

α(i) = [p̃H(i)d̃(i)]−1(λ − η)p̃H(i)g̃(i − 1) (0 ≤ η ≤ 0.5)ṽ(i) = ṽ(i − 1) + α(i)p̃(i)g̃(i) = (1 − λ)ã(θ1) + λg̃(i − 1) − [r̃(i)r̃H(i) + ξI2M ]ṽ(i − 1) − α(i)d̃(i)β(i) = [g̃H(i − 1)g̃(i − 1)]−1[g̃(i) − g̃(i − 1)]H g̃(i)p̃(i + 1) = g̃(i) + β(i)p̃(i)w̃(i) = [ãH(θ1)ṽ(i)]

−1γṽ(i)end

rearranging the terms, we have that

(λ− 0.5)E

{p̃H(i)g̃(i− 1)

}

E


} ≤ E {α(i)} ≤ λE

{p̃H(i)g̃(i− 1)

}

E


} . (3.47)

From (3.47), the inequality is satisfied if α(i) is computed by

α(i) = (λ− η) p̃H(i)g̃(i− 1)

p̃H(i) ˆ̃R(i)p̃(i), (3.48)

where 0 ≤ η ≤ 0.5 to fulfill the convergence criterion. The step size β(i) is computedaccording to the Polak-Ribiere approach [30] to avoid the reset procedure [8] and is

stated as

β(i) =[g̃(i) − g̃(i− 1)]H g̃(i)

g̃(i− 1)H g̃(i− 1) . (3.49)

The proposed WL-MCG algorithm for widely-linear adaptive beamforming is sum-

marized in Table 3.3. It only performs one iteration of the CG algorithm per snapshot,

saving significant computational efforts. The diagonal loading factor ξ in the algorithm

is again chosen to be ξ = 10 · σ2n [27].

3.7 Computational complexity

The computational effort of the WL processing is one of the most important aspects

to take into account when developing novel WL adaptive beamforming algorithms.

To this end, we evaluate the computational complexity of the proposed WL Krylov-

based beamforming algorithms and compare them to their linear counterparts. For



comparison purposes, we first focus on the complexity of the linear algorithms, and

then consider the complexity of the WL algorithms.

3.7.1 Complexity of LCMV algorithms

The computational cost, shown in Table 3.4, is measured in terms of the arithmetical

operations, whereM is the number of sensor elements andK is the number of iterations.

The expressions in Table 3.4 are obtained from the work in [23]. It is evident that the

complexity of the AVF and the CG algorithm depends on the iteration number K.

Figure 3.5 illustrates the computational cost as a function of the number of sensors M .

The iteration numbers for the AVF and the CG method are chosen to be K = 4 and

K = 5 respectively, as these numbers are used in the following performance evaluation.

Comparing the cost of the linear Krylov-based algorithms to the RLS method, the AVF

technique clearly exceeds the computational requirements of the RLS method, whereas

the CG algorithm demands a slightly higher cost. The aforementioned trade-off of

the CG method only becomes apparent for a smaller number of iterations. However,

the MCG algorithm provides a lower computational burden than the RLS technique,

especially for a large number of sensors M , but still requires a higher cost than the

LMS algorithm.

10 20 30 40 5010

0

101

102

103

104

105

Num

ber

of additio

ns

Number of sensors M10 20 30 40 50

100

101

102

103

104

105

Num

ber

of m

ultip

lications

Number of sensors M

LMS

WL−LMS

RLS

WL−RLS

AVF, K = 4

WL−AVF, K = 2

CCG, K = 5

WL−CCG, K = 3

MCG

WL−MCG

Fig. 3.5: Complexity in terms of arithmetic operations versus the sensors M .



Tab. 3.4: Computational complexity of LCMV beamforming algorithms

Algorithm Additions Multiplications

LMS [5] 4M − 2 4M + 3RLS [5] 4M2 − M − 1 5M2 + 5M − 1AVF [9] K(4M2 + M − 2) + 5M2 − M − 1 K(5M2 + 3M) + 8M2 + 2MCG [8] K(M2 + 4M − 2) + 2M2 − 1 K(M2 + 4M + 1) + 3M2 + 3MCG [8] 2M2 + 7M − 3 3M2 + 9M + 4

Tab. 3.5: Computational complexity of WL-CMV beamforming algorithms

Algorithm Additions Multiplications

WL-LMS [21] 8M − 2 8M + 3WL-RLS [22] 16M2 − 2M − 1 20M2 + 10M − 1WL-AVF K(16M2 + 2M − 2) + 20M2 − 2 ∗ M − 1 K(20M2 + 6M) + 32M2 + 4MWL-CG K(4M2 + 8M − 2) + 8M2 − 1 K(4M2 + 8M + 1) + 12M2 + 3WL-MCG 8M2 + 14M − 3 12M2 + 18M + 4

3.7.2 Complexity of WL-CMV algorithms

The computational complexity for the WL beamforming algorithms is depicted in Table

3.5. Due to the WL processing, the analyzed algorithms developed for non-circular

sources consider the additional information contained in the augmented steering vector,

which can be interpreted as an array composed of twice as many sensor elements M .

Thus, the computational cost of all the WL beamforming methods increases by nearly

the same amount, complying with the structure of the augmented covariance matrix

R̃. As a result, the same observations for the linear methods also apply to the WL

case. According to Figure 3.5, where the number of iterations for the WL-AVF and

the WL-CG techniques is now K = 2 and K = 3 respectively, the gap between the

cost of the WL-RLS and the WL-CG is substantially decreased. The complexity of

some of the considered algorithms might be further reduced by using symmetries and

individual properties. However, these complexity reductions are beyond the scope of

this work and are left for future investigations.



3.8 Simulation results

In this section, the output SINR performance of the proposed WL Krylov-based al-

gorithms in stationary and non-stationary scenarios is evaluated and compared to the

WL-RLS, the WL-LMS algorithm, and their linear counterparts. For the simulations,

we employ a ULA consisting of M = 24 sensor elements whose interelement spacing

equals half a signal wavelength. Among d sources in the system, the DOA of the desired

user with signal power σ21 = 1 is θ1 = 0◦, which is known beforehand by the receiver. All

the signals are assumed to be binary phase-shift keying (BPSK)-modulated, and γ = 1

is set to satisfy the condition for the MVDR leading to a maximum achievable SINR

performance. The diagonal loading factor is ξ = 10 · σ2n and each curve is obtained byaveraging a total of 1000 runs. Notice that in the simulations the number of iterations

K for the WL-AVF and the WL-CG algorithm is fixed for all the realizations. The first

part of the computer simulations deals with the performance in stationary scenarios

and the second part is concerned with the performance assessment in non-stationary

scenarios.

3.8.1 Performance in stationary scenarios

In the first experiment, we examine the SINR performance of the proposed WL adaptive

beamforming algorithms in a stationary system. Assuming that there are d = 15 users

in the scenario, the d − 1 = 14 interfering signals impinge on the sensor array fromthe directions (±10◦ · [1, . . . , (d − 1)/2]) with an input SNR = 10 dB and a signal-to-interference ratio (SIR) of −20 dB. Due to the stationarity of the system, the forgettingfactor is set to be λ = 1.

Figure 3.6 shows the output SINR performance as a function of the number of

snapshots N . The number of iterations K for the AVF algorithm and for the WL-

AVF algorithm were found to be K = 4 and K = 2 and the iteration number for the

CG algorithm and the WL-CG algorithm is K = 3 and K = 5, respectively. The

amount of iterations was optimized to yield the best output performance in terms

of the convergence speed. The lower number of iterations for the WL version of the

AVF and the CG algorithm can be explained by the fact that the augmented weight

vector provides additional information and reaches the optimum solution with fewer

iterations. It is apparent from Figure 3.6 that all the WL beamforming algorithms

provide substantial improvements over the linear techniques in general. Specifically,

the proposed WL-AVF algorithm outperforms the adaptive WL-LMS and the WL-RLS

in terms of both the convergence and the steady state. The WL-CG method and the



0 200 400 600 800 1000

−2

0

2

4

6

8

10

12

14

Outp

ut S

INR

(dB

)

Number of snapshots N

LMS

WL−LMS

RLS

WL−RLS

AVF, K = 4

WL AVF, K = 2

CG, K = 5

WL−CG, K = 3

MCG

WL−MCG

SINRmax

LCMV

SINRmax

WL−CMV

Fig. 3.6: Output SINR versus the number of snapshots N in a stationary scenario withM = 24, d = 15, SNR = 10 dB, SIR = −20 dB, λ = 1.

0 5 10 15 200

5

10

15

20

25

Outp

ut S

INR

(dB

)

SNR (dB)

LMS

WL−LMS

RLS

WL−RLS

AVF, K = 4

WL AVF, K = 2

CG, K = 5

WL−CG, K = 3

MCG

WL−MCG

SINRmax

LCMV

SINRmax

WL−CMV

Fig. 3.7: Output SINR versus the input SNR in a stationary scenario with M = 24,d = 15, N = 1000, SIR = −20 dB, λ = 1.



WL-MCG algorithm perform close to the level of the WL-RLS algorithm and outclass

the WL-LMS technique. However, as discussed in the previous section, the WL-MCG

algorithm significantly reduces the complexity of the WL-CG method. As seen for the

SINRWL-CMVmax compared to the SINRLCMVmax , by exploiting the additional information in

the presence of non-circular sources, a higher gain can be achieved. This is due to the

WL processing, where the augmented steering vector can be interpreted as a virtual

array with twice as many sensors.

The SINR performance as a function of the SNR is illustrated in Figure 3.7, where

the number of snapshots is fixed to N = 1000. As expected, the SINR increases with

the SNR, and for the SINRWL-CMVmax the additional gain is expressed by shifting the curve

for the SINRLCMVmax to a higher SINR. The proposed WL algorithms performs similarly

to the results in Figure 3.6. The WL-AVF method outperforms the WL-LMS and the

WL-RLS algorithm, whereas the WL-CG and the WL-MCG techniques almost reach

the level of the WL-RLS algorithm.

3.8.2 Performance in non-stationary scenarios

In the second experiment, the SINR performance in a non-stationary scenario is an-

alyzed and compared to the WL-LMS and the WL-RLS. The system starts with the

same scenario as in Figure 3.6, but with different power levels of the interfering signals,

0 500 1000 1500 2000−2

0

2

4

6

8

10

12

14

Ou

tpu

t S

INR

(d

B)

Number of snapshots N

LMS

WL−LMS

RLS

WL−RLS

AVF, K = 4

WL AVF, K = 2

CG, K = 5

WL−CG, K = 3

MCG

WL−MCG

Fig. 3.8: Output SINR versus the number of snapshots N in a non-stationary scenariowith M = 24, d1 = 15, d2 = 12, SNR = 10 dB, λ = 0.998.



namely 8 interferers with a power of 20 dB, 4 with 10 dB and 2 with 5 dB above the

level of the SOI. At the 1000th snapshot, 4 users with a signal power of 20 dB and

one user with 10 dB above the SOI leave the system, and 2 users with a power of

20 dB and 5 dB above the SOI enter the system. As can be seen in Figure 3.8, the

proposed WL-CG and the WL-MCG quickly adapt to the changed system and exhibit

the convergence speed and the steady-state level of the WL-RLS method. Thus, the

WL-CG and the WL-MCG approaches provide a better performance after a change in

the environment. Nevertheless, the same conclusion can not be drawn for the proposed

WL-AVF algorithm. After the changing event, it barely exceeds the convergence speed

of the WL-LMS method and only attains the level of the other algorithms after requir-

ing a significant number of data samples. This behavior is due to the relatively low

number of iterations of the AVF algorithm per data update. However, a larger iteration

number would decrease the performance in a stationary system. In consequence of the

fixed number of iterations K for the experiments, it is evident that the performance

can be improved by applying an automatic scheme to adaptively adjust K.

3.9 Conclusions

In this chapter, an extension of the Krylov subspace-based algorithms employing the

AVF, the CG and the MCG methods for adaptive beamforming to the WL process-

ing is developed, resulting in the WL-AVF, the WL-CG and the WL-MCG algorithm.

After exploring the performance advantages associated with the WL processing in the

face of implying a loss of the computational efficiency, the WL-CMV optimization

criterion is introduced to update the augmented beamforming weight vector. Then

by assuming strictly non-circular signals in the system, the general framework of the

three proposed adaptive algorithms is discussed and their detailed derivation presented.

The developed algorithms fully exploit the second-order statistics of the strictly non-

circular signals and avoid the computation of the inverse of the augmented covariance

matrix. Whereas the WL-AVF algorithm applies augmented auxiliary vectors within

a number of iterations to adapt the augmented weight vector at each data sample, the

WL-CG resorts to augmented residual vectors. The key benefit of the WL-MCG is

the fact that it only performs one iteration per data snapshot. To assess the compu-

tational requirements, an analysis of the complexity is conducted, which demonstrates

that the same computational behavior in the linear case also holds for the WL pro-

cessing. Specifically, the WL-AVF algorithm requires the highest complexity and the

WL-MCG the lowest cost. Furthermore, the SINR performance of these algorithms is

evaluated by computer simulations for stationary and non-stationary scenarios, where



the number of iterations for the WL-AVF and the WL-CG method is optimized. It is

illustrated that the WL processing provides significant performance improvements over

the linear processing. In stationary scenarios, the WL-AVF algorithm outperforms the

WL-RLS method, and the WL-CG and the WL-MCG algorithm perform close to the

WL-RLS technique. Considering non-stationary scenarios, the WL-CG and the WL-

MCG method attain the performance of the WL-RLS approach after a change in the

system; however, the WL-AVF achieves a slower adaptation due to the lower number of

iterations. This motivates an automatic scheme for adaptively switching the iteration

number.


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